Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

7.8 Second Moments of Areas 187

Now let us expand this problem to include more small area elements, as shown in Figure 7.23.

The area moment of inertia for the system of discrete areas shown about thez–zaxis

is now

(7.9)

Similarly, we can obtain the second moment of area for a cross-sectional area such as a rectangle

or a circle by summing the area moment of inertia of all the little area elements that

make up the cross section. As you take calculus classes, you will learn that you can use integrals

instead of summing ther

2

Aterms to evaluate the area moment of inertia of a continuous

cross-sectional area. After all, the integral sign, , is nothing but a big “S” sign, indicating

summation.

(7.10)

Also note that the reason this property of an area is called “second moment of area” is that

the definition contains the product ofdistance squaredand an area, hence the name “second

moment of area.” In Chapter 10, we will discuss the proper definition of a moment and how

it is used in relation to the tendency of unbalanced forces to rotate things. As you will learn later,

the magnitude of a moment of a force about a point is determined by the product of the per-

pendiculardistancefrom the point about which the moment is taken to the line of action of the

force and the magnitude of that force. You have to pay attention to what is meant by “a moment

of a force about a point or an axis” and the way the termmomentis incorporated into the name

“the second moment of area” or “the area moment of inertia.” Because the distance term is

multiplied by another quantity (area), the word “moment” appears in the name of this prop-

erty of an area.

You can obtain the area moment of inertia of any geometric shape by performing the inte-

gration given by Equation (7.10). You will be able to perform the integration and better under-

stand what these terms mean in another semester or two. Keep a close watch for them in the

upcoming semesters. For now, we will give you the formulas for area moment of inertia with-

out proof. Examples of area moment of inertia formulas for some common geometric shapes

are given next.

Izzr

2

dA

Izzr

2

1 A 1 r^

2

2 A 2 r^

2

3 A 3

r 1 A 1

z

z

r 2 A 2

■Figure 7.23 r 3 A 3

Second moment of area for three

small area elements.

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